Various statistical quantities are used to infer causal relationships [1]}.
In pigeon flocks, for example, time-delayed correlation between
the orientation of individuals at one time instance and the orientation of
others at previous times reveals a hierarchical leadership structure and also
provides a method to quantify the timescale of influence [2]}. In
such a case, the motion of one pigeon is correlated with the past motion of
another. Granger causality [3]} is seen as an improvement to
time-delayed correlation as it quantifies the predictability of the current
state of a variable based on knowledge of a variable at a previous time.
Time-delayed correlation and Granger causality both assume linear relationships
between variables, though. This generally does not hold. More recent studies
argued that information-theoretic quantities—transfer entropy, time-delayed
mutual information, and causation entropy—are superior when quantifying influence
since they naturally accommodate the highly nonlinear nature of
multi-agent systems [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [11]}, [15]}, [16]}.
